Fourier transform recording with random phase shifting

ABSTRACT

In making a record of the exact Fourier transform of an array of beams of electromagnetic radiation, the phase of each of a substantial fraction of the beams is shifted by a constant amount before recording the transform.

vllllcu dlillcs rate! [H] [72] Inventor Christoph B. Burckhardt [56]References Clled Berkeley Heights NJ. 211 Appl. No. 868,485 OTHERREFERENCES [22] Filed 0342,1969 Murata et al., Japanese Jour. of AppliedPhysics, Vol [45] at Sept. 197] March 1968, pp. 301- 302 (copy in350/35) Pennington, IBM Technical Disclosure Bulletin, Vol. ll, No. 3,Aug. 1968 pp. 322- 323 (copy in 350/3.5)

Leith et al., Applied Optics, Vol. 7, No. 10, Oct. 1968 pp.

[ 73] Assignee Bell Telephone Laboratorles, Incorporated Murray Hill,Berkeley Heights, NJ.

\ 2085- 2089 (copy in 350/35) 54 FOURIER TRANSFORM RECORDING WITHPrimary Examiner-David Schonbcre RANDOM PHASE SHIPPING AssistantExaminer- Ronald J. Stern 9 Claims, 6 Drawing Fi Attorneys-R. J.Guenther and Arthur J Torsiglieri [52] US. Cl 350/35,

96/27 H ABSTRACT: In making a record of the exact Fourier trans- [5l]Int. Cl ..G02b 27/ 0 form of an array of beams of electromagneticradiation, the {50] surch 350/35; phase of each ofa substantial fractionof the beams is shifted 96/27 R, 27 E, 27 H; 346/107, 1 45.2 by aconstant amount before recording the transform.

LIGHT SOURCE DEFLECTING SYSTEM PATENTED SEP 1 41971 SHEET 1 [IF 3 FIG. 2

/N VENOP c. B. BURC/(HARDT A TTOFPNEV ATENTEU SEP] 41% 3604.778

sum 3 OF 3 DEFLECTI NG LIGHT SOURCE MAN IPULATING MEANS 616 FOURIERTRANSFORM RECORDING wrrn RANDOM PHASE smrrmc BACKGROUND OF THE INVENTIONThis concerns Fourier transform recording and in particular therecording of the Fourier transform of an array of beams ofelectromagnetic radiation.

As is well known, the Fourier transform of the amplitude and phasedistribution of radiation at a first location in a beam of radiation issimply an amplitude and phase distribution at a different location inthe beam that is a mathematical Fourier transform of the firstdistribution. Such a Fourier transform may be made in several ways. Forexample, with optical techniques, if the amplitude and phasedistribution that is to be transformed is located in the front focalplane of a lens, then the Fourier transform of this distribution isformed in the rear focal plane, which is also called the Fouriertransform plane. Alternatively, if the amplitude and phase distributionthat is to be transformed is located in a converging light beam fonnedby a beam of parallel light that is incident on a converging lens, thenthe Fourier transform of this distribution multiplied by a sphericalphase factor is formed in the focal plane of the converging lens.Further details about Fourier transforms and about optical Fouriertransforms may be found in R. Bracewell's The Fourier Transform and itsApplications" (McGraw-Hill 1965); .l. W. Goodmans Introduction toFourier Optics" (McGraw-Hill, 1968); and S. G. Lipson and H. LipsonsOptical Physics" (Cambridge University Press, 1969).

A recently developed application of Fourier transform techniques hasbeen their use in the fonnation of hologram memories. As is well known,a hologram is a record of the interference pattern produced by theinterference of a coherent reference-beam with a phase-relatedinfonnation-bearing beam from am object. When a hologram is illuminatedwith one of the two beams used in forming it, the other beam isdiffracted from it. ln particular, when a hologram is illuminated withthe reference-beam used in forming it, the informationbearing beam isreconstructed; and an image of the object that originally formed theinformation-beam can be detected. Typically, this image can be eitherreal or virtual depending on how the reference beam illuminates thehologram.

These properties can be used to advantage in forming a hologram memorythat is in essence a record of the interference between a reference-beamand a large array of minute information-bearing beams derived from anappropriate object. In one such hologram memory detailed by V. A. Vitolsin his paper Hologram Memory for Storing Digital Data" at page 1581 ofthe IBM TECHNICAL DISCLOSURE BUL- LETlN, VoL, 8, No. 11 Apr. 1966) andby F. M. Smits and L. E. Gallagher in their paper Considerations for aSemipermanent Optical Memory at page 1267 of the BELL SYSTEM TECHNICALJOURNAL, Vol. 46, No. 6 (July-Aug, 1967), the information-bearing beamsare typically fonned by illuminating a mask-bearing regularly spacedindex points or bit positions at which are selectively located indiciarepresenting bits or units of digital data. Illustratively, the presenceat an index point or bit position of an aperture in the otherwise opaquedata mask signifies a l bit while the absence of such an aperture at abit position signifies a bit; and, consequently, illumination of theopaque mask produces an array of points of light or minuteinformation-bearing beams at the apertures in the mask. Because the beamthat illuminates the data mask is coherent and phase-related to thereferencebeam, the informationbearing beams derived from the mask arelikewise coherent and phase-related to the referencebeam; and theinterference of these beams is recorded as a hologram on a suitablemedium.

Because very little space is required on the recording medium to store ahologram of as many as several thousand bits of digital data, it ispossible to store on different areas of the same recording mediumdifferent holograms of different groups or pages of digital data. Onesimply exposes one area of the recording medium to one data page, thensubstitutes another data page for the first, lines up. an unexposedportion of the recording medium with the new data page and exposes thatpreviously unexposed portion to the new page. The result of such aprocedure is to form on the recording medium an array of holograms, eachof which is a recording of a page of digital data,

To read the memory, one hologram at a time is illuminated with thereference-beam to reconstruct the original information-bearing beams insuch a way that they form a real image comprised of an array of spots oflight representative of the array of apertures in the opaque mask usedto form the hologram. Appropriate readout devices such as an array ofphotodetectors are then used to sense the presence or absence ofparticular spots of light or information-bearing beams in the realimage.

Such a system as that described above has several attractive features. Ahologram inherently has optical properties similar to those of a lens.Hence separate lenses are not required to image the contents of thehologram memory onto the array of photodetectors. Second, because theresolution obtainable in a unity-magnification-imaging.magnification-imaging situation is close to the maximum theoreticallimits, each light spot that is imaged onto a photodetector is as smalland as intense as possible. Lastly, the capacity and speed of thehologram memory system are quite high. in their article, Smits andGallagher demonstrate that the capacity of the memory is in excess ofl00-million bits if the data is stored in the form of approximately10,000 holograms each containing approximately 10,000 bits of data.Moreover, the access time to any one hologram can be less than tenmicroseconds lOpsec.

While the system described above is attractive, it is also desirable tomake the system relatively insensitive to blemishes or dust on thehologram-recording medium so that a small blemish or dust particle onthe hologram memory cannot obscure or change a bit of digital data.Clearly, this advantage canbe achieved if the information about eachinformation-bearing beam is stored throughout one hologram, rather thanin a small area; and it can be shown that just such storage can beachieved by recording the information-bearingbeams in the form of ahologram of the Fourier transform of the information-bearing beams.However, while it may be fairly easy to form optically the Fouriertransform of a distribution of radiation, it is not always so easy torecord a Fourier transform of an ordered. array of light beams.Specifically, an ordered array of light beams such as those produced bythe apertures of the data mask used in a hologram memory is describedmathematically as havingan amplitude and phase distribution that iscomprised of an ordered array of sharp, highamplitude peaks or spikes ofconstant phase; and the Fourier transform of such an amplitudeand phasedistribution is a second ordered array of sharp, high-amplitude andhighintensity peaks or spikes in which each spike in the first arraycontributes something to theamplitude and intensity of every one of thespikes in the second array. The enormous difference in intensity,however, between the light in the Fourier transform spikes and that inthe surrounding regions may make it difficult, if notimpossible, torecord the Fourier transform in the linear region of response in therecording medium. And, consequently, in applications such as holographywhere a linear-recording response is desired in order to avoid.distortions, some way must be found to distribute the Fourier transformlight more evenly over the Fourier transform plane.

One method that has been proposed is that the hologram. recording bemade in some plane other than the exact Fourier transform plane.However, while such a recording technique can result in more even lightdistribution, the optical system that must be used with such a method isconsiderably more complicated and more expensive than the optical systemthat is used to form a hologram in the exact Fourier transform plane.Another method that has been proposed for attaining a more evenlightdistribution in the Fourier. transform plane is that the phase of thelight beams incident on the Fourier transforming lens be shifted atrandom with respect to each other because it can be shown that if thephase of the light beams is so shifted then the amplitude distributionin the Fourier transform plane is indeed substantially uniform. However,unless the phase shifi within each beam is a constant, there may resultan inferior reconstruction of the original beams. And, indeed, the lightlevel in particular reconstructed beams may even be reduced so low thatthe presence of the beam cannot be detected by the readout device.Obviously, where each beam represents a binary bit that forms part of aunit of information, it is not practical to run the risk of having bitsaltered by indiscriminate phase-shifting.

SUMMARY OF THE INVENTION Accordingly, it is an object of this inventionto improve the recording of Fourier transforms and in particular therecording of Fourier transforms of ordered arrays of beams of radiation.

And it is a further object of this invention to improve the recording ofFourier transforms in which phase-shifting is used to distribute theamplitude in the transform more evenly over the Fourier transform plane.

In an illustrative embodiment of my invention, these and other objectsare achieved in recording the Fourier transform by shifting the phase ofapproximately one-half the beams of the array to be transformed byapproximately 180 before the transform is completed. Thus, if the arrayof beams is an array of light beams emanating from the apertures in anotherwise opaque mask that is located in a converging light beam formedby a beam of parallel light incident on a converging lens, the phase ofapproximately one-half these light beams is shifted approximately 180before the light is incident on the rear focal plane of the lens.Advantageously, this phase shift is effected by a second mask locatednext to the aperture mask.

BRIEF DESCRIPTION OF THE DRAWING These and other elements, features andobjects of my invention will be more readily understood from thefollowing detailed description of the invention taken in conjunctionwith the following drawing in which:

FIG. I shows an illustrative embodiment of my invention;

FIG. 2 shows a schematic view of the data mask used with theillustrative embodiment of my invention;

FIG. 3 shows a schematic view of the phase mask used with theillustrative embodiment of my invention;

F IG. 4 shows a schematic view of the relation between the data mask andthe phase mask in the illustrative embodiment of my invention;

FIG. 5 shows illustrative apparatus used to reconstruct information froma hologram formed with the illustrative embodiment of my invention; and

FIG. 6 shows illustrative apparatus for aligning the data mask and thephase mask.

DETAILED DESCRIPTION OF THE DRAWING Shown in FIG. 1 is illustrativeapparatus used in the practice of my invention. The apparatus iscomprised of a light source 5, a deflecting system 7, a Fouriertransforming lens 11, a data mask 14 that is comprised of an opaquemedium in which there are small transparent apertures, a phase mask 15,and a photosensitive recording medium 19 that is located in the rearfocal plane, which is also called the Fourier transform plane, of lens11. Light source 5 is preferably a high-power laser such as the YAG: Ndlaser, frequency doubled to produce a watt of green light. Many otherlasers are also available. Deflecting system 7 is typically a pair ofcascaded acousto-optic light deflectors oriented to produce lightdeflection in two orthogonal directions and appropriate lens elementsarranged to render parallel the different beam paths down which thelight may be deflected. In addition, deflecting system 7 also containsthe necessary optical elements, such as beam splitters and reflectors,to split the light from source 5 into two coherent light beams having aconstant phase relation and to 7 direct these two beams onto the sameportion of recording medium 19 at an angle with respect to each other.Consequently, when the device of FIG. 1 is operated, an interferencepattern is formed on a particular portion of recording medium 19, theprecise portion depending on where the two beams were deflected theacousto-optic deflectors. Because light source 5 and deflecting system 7are well known in the art, they are represented in FIG. 1 in blockdiagrams; and further discussion of them will be limited to an analysisof a representative illuminating beam 9 and reference-beam 18 producedby them. 7

As described above and as shown in FIG. 2, data mask 14 bears an arrayof index points or bit positions 24 that are representative of digitaldata. Typically, these bit positions are arranged in a square array sothat adjacent bit positions are equidistant, their centers beingseparated by a distance d. I]- lustratively, at those bit positionswhere a 1" bit is to be represented, there is a transparent aperture inmask 14 while at those bit positions where a 0" bit is to be representedthere is no aperture. Thus, in the upper left-hand corner of the maskshown in FIG. 2 the first three bit positions in the first row aretransparent, signifying 1 bits, while the first three bit positions inthe second row are opaque, signifying 0" bits. There are numerous waysof representing the data on mask 14. As suggested above, data can berepresented simply by holes in an otherwise opaque mask. Alternatively,more sophisticated mask systems using electrically switched Pockels orKerr cells and polarizers can be used as described in U.S. Pat. No.3,530,442, issued to R. J. Collier and L. H. Lin, and assigned to BellTelephone Laboratories, Inc. The output of either representation of datais the same in that each provides an array of light beams from thosepositions on the mask registering one of the binary bits and no lightbeams from the positions registering the other binary bit.

As shown in FIG. 3, phase mask 15 is comprised of an array oftransparent squares 25, the edges of which have a length d that is thesame as the spacing between the centers of the equidistant bitpositions24 of mask 14. As indicated by the designation 180 in some of thesquares 25 of phase mask 15, some of the squares shift the phase of thelight transmitted through them by 180 with respect to the phase of thelight that goes through the squares labeled 0. As will be detailedbelow, approximately 50 percent of the squares are fabricated to shiftthe phase of incident light by and these squares are distributedrandomly throughout phase mask 15.

In the apparatus of FIG. I data mask 14 and phase mask 15 are aligned sothat light from each bet position 24 of data mask 14 goes through onlyone square 25 of phase mask 15, thereby establishing a one-to-onecorrespondence between the bit positions of mask 14 and thephase-shifting areas of mask 15. As shown in FIG. 4, the combination ofthe two masks then looks like a data mask where approximately half thebit positions have associated with them a phase of 0 and the remainingbit positions have a phase of Consequently, there is a probability ofapproximately one-half that a light beam from a bit position 24 willhave its phase shifted by 180 by the particular square 25 of mask 15through which it passes.

To record a Fourier transform hologram on a particular portion ofrecording medium 19, coherent light is directed from light source 5 todeflecting system 7 where it is deflected and fanned into theilluminating beam 9 and the reference beam 18. Illuminating beam 9,which is preferably a beam of substantially parallel light such as wouldbe derived from a source of light an infinite distance away, is thendirected onto Fourier transforming lens 11 which focuses it throughmasks l4 and 15 onto the desired portion of recording medium 19 locatedin the rear focal plane, or Fourier transform plane, of lens 11. Asindicated above, mask 14 represents an array of binary data in the formof the presence or absence of light transmitting apertures at an orderedarray of index points or bit positions on the mask. Consequently, whenmask 14 is illulight beam of the array transits only one of thephase-shifting squares of mask and because the 180 phaseshifting squaresare distributed randomly on the mask, the

Once the data has been changed and the appropriate voltages applied tothe deflectors, light is directed from light source 5 to deflectorsystem 7 where it is deflected and formed into an illuminating beam anda reference-beam. Again, the illuminating beam is directed through lens11 to data mask 14 and phase mask 15 where it is formed intotion-bearing beams, approximately half of which have a phase that isapproximately 180 different from the phase of the remaining beams. Thesebeams then interfere with the reference-beam on the selected portion ofrecording medium 19, and the resulting interference pattern is likewiserecorded as a hologram on medium 19.

To store a large number of holograms on one recording medium, it isdesirable to record each hologram on only a very data mask. the totalcapacity of a recording medium is approximately l0bits.

To read any one of the holograms stored on medium 19 in distance betweendata mask 14 and recording medium 19 in F IG. l. Advantageously, lightsource 505 and deflecting system 507 are similar to light source 5 anddeflecting system 7 used in FIG. 1 to record paratus to read aparticular hologram, appropriate voltages be read.

As is well known in holography, a beam that is antiparallel to thereference-beam is the conjugate of the reference-beam travelling in adirection opposite to that of the referenceopposite to the direction ofthe information-bearing beam that formed the hologram and reconstructs areal image of the object that formed the information-bearing beam at thesame distance from the hologram as the object was when the hologram wasformed. Consequently, when an antiparallel reconstructing beam 518 isincident on the hologram on medium 519, at least some of it isdiffracted by the hologram to reconstruct a real image 514 of the datamask that formed information-bearing beams 16 of FIG. 1. As indicated inFIG. 5, this Array 525 is illustratively comprised of an array of lightsensitive photodiodes 526 such as that described in the aforementionedarticle by Smits and Gallagher. The array is in the plane of image 514and contains as many photodiodes as there can be spots of light in theimage, one photodiode being lined up at be transferred into an electricsignal; and this signal can be stored, for example, in a flip-flop in abuffer memory.

In a similar fashion, any of the other holograms recorded on medium 519can be read by a reconstructing beam that is anstructed images arecoincident; and each image can therefore be read by one diode array,array 514. Again, if each hologram is recorded on only a small area ofmedium 519, it may illuminated by the reconstructing beam.

The effect of the random phase shifting of the information-bearing beamshas been analyzed mathematically in a fashion analogous to the analysisof the random telegraph wave at pages 326 and 327 of S. O. Rice's paper,"Mathematical Analysis of Random Noise," BELL SYSTEM TECHNI- CALJOURNAL, Vol. 23, p, 282 July 1944). As a result of such analysis, ithas been found that the intensity distribution in the Fourier transformplane of an array of light beams whose phase has been shifted inaccordance with my invention is proportional to the intensity of theFourier transform of one pattern, and just such a pattern ing myinvention.

The effect of inaccurate phase-shifts has also been investigated and hasbeen found to create a pattern in which an array of sharp,high-intensity spikes are superimposed on the Airy pattern. A fonnulahas been derived for determining how much power P,goes into the array ofsharp, high-intensity spikes as a result of an inaccurate phase shift.Specifically, P,= P sin (8/2l the holograms. ln operating the apwhere Pis the total power and is the deviation frgrn an accurate phase shift of180. Thus, for 5=10, P ,=0.76% of P; for 8=20, P,=3% of P; for 6=40, P,=11.7% of P: and for 8=50, P,= 17.9% of P. Obviously, any phase shifttransfers power from the ordered array of high-intensity spikes to theAiry pattern and a phase shift of 180 in half the light beams willtransfer all th e ptlwt=: r to tl1 e Airy pattern. Complete transfer ofpower, while preferable, is not generally required; and to varyingdegrees other phase shifts are acceptable depending on the particularapplication of my invention. When my invention is used in recordingForier transform holograms of ordered arrays of light beams, a lightintensity pattern is acceptable in which 10% of the power is in theordered array of high-intensity spikes; and accordingly, the phase shiftthat is used can range between approximately 140 and 220.

Similarly, it is not necessary to shift the phase of exactly one-halfthe light beams or to use a pattern that is completely random. Indeed,even in random patterns small regularities will appear. Again, theprecise number of beams that must be phase-shifted and the degree ofrandomness required will vary depending on the intensity distributiondesired in a particular application. For my purposes, it appearsacceptable to shift the phase of approximately 40 percent to 60 percentof the light beams that are being Fourier transformed; and,consequently, the probability that a particular beam will bephaseshifted can range from two-fifths (0.40) to three-fifths (0.60) Atthe extremes, the percentage of beams that are phase shifted might rangefrom about 33% percent to about 66% percent, and the probability of abeam being phase-shifted might range from about one-third to abouttwo-thirds.

Phase mask was fabricated quite easily using standard photoresist andetching techniques. With the aid of a random number generator. I firstformed a pattern comprised of approximately equal numbers of opaque andtransparent squares randomly interspersed. This pattern was then reducedphotographically to the scale of the data mask and was used to etchglass according to a procedure detailed at pages 16 and 17 of Kodakpublication P-91 Applications Data for Kodak Photosensitive Resists.Briefly, glass slides coated with a 2,000 Angstrom thick layer ofevaporated silver were covered with KTFR photo-resist, the pattern ofopaque and transparent squares was contact printed on the photoresist,and the photoresist was developed.

The silver film in the unexposed areas was then etched away inconcentrated Farmer's reducer with no agitation. The particular reducersolution used was 1 part of solution A, 4 parts of solution B and 10parts of distilled water where solution A was composed of 37.5 grams ofPotassium Ferricyanide in enough distilled water to make 500 cc. ofsolution and solution B was composed of 480 grams of Sodium Thiosulfatein enough distilled water to make 2,000 cc. of solution. Somewhat morethan 30 minutes was needed to etch the silver film with the solutionbeing changed every 15 minutes. For convenience, the etching was done ina transparent Petri dish placed on top of an illuminator so that theetching could be observed and terminated as soon as the silver under theunexposed areas of the photoresist was completely etched. Afterwards theback side of the glass slide was coated with paraffin, and the exposedglass was etched in a solution of 10 cc. of hydrofluoric acid (49percent) and 200 cc. of distilled water. Next, the photoresist was wipedoff with xylene, and the remaining silver film was etched off withdiluted nitric acid. The phase-shift of the etched glass slide was thenmeasured with a Mach-Zender type interferometer. By trial and error itwas found that the time needed to etch glass to a depth corresponding toa 180 phase-shift at a wavelength of 6,328 Angstroms was 2 minutes.

As emphasized above. it is also necessary that the phase mask be alignedwith the data mask so that light from each bit position on the data maskcan pass through only one of the phase-shifting squares of the phasemask. This alignment is readily accomplished with the apparatus of FIG.6. In this system there is shown a lens 611, a data mask 614. a phasemask 615 that is to be aligned with mask 614, manipulating means 616 fortranslating and rotating mask 615, an aperture 617 in an otherwiseopaque medium 618 located in the focal plane or Fourier transform planeof lens 611, a second lens 620 positioned so that aperture 617 is in itsfront focal plane, and a screen 621. When lens 611 is illuminated by abeam of parallel light from an appropriate source. not shown, itconverges the beam through data mask 614 and phase mask 615 to focus onaperture 617 in medium 618. The light that transits aperture 617 is thenformed by lens 620 into an image of the data mask thatcan be observed onscreen 621. Before alignment of the phase mask with the data mask, apattern of data spots with reduced intensity can be observed in thisimage because any light from the data mask that is incident partly on asquare with a 0 phase-shift and partly on a square with a 180phase-shift is deflected away from aperture 617 and does not contributeto the formation of the image on screen 621. Thus, to align the twomasks, I simply rotate and translate the phase mask with themanipulating means 616 until the image of data mask 614 on screen 621 isuniformly bright, indicating that the phase is constant over the area ofevery data spot and that the phase mask and the data mask are thereforealigned.

As is obvious from the foregoing, there are numerous ways to practice myinvention and the preceding description is only meant to be illustrativeof my invention. INn FIG. I. any light source can be used that iscapable of producing a light beam that is coherent enough to be used informing a hologram. Other methods are available for forming the Fouriertransform of a light beam such as the methods known in the art anddescribed in US. Pat. No. 3,533,676, issued to L. H. Lin, Ser. No. andassigned to Bell Telephone Laboratories, Inc. Similarly, there arenumerous ways of forming the data mask and of recording the hologram ofthe light beams from the data mask. Some of these methods have beendetailed above, and there are others well known to those of ordinaryskill in the art. In addition, there are alternative ways of arrangingthe phase mask with the data mask because it is only necessary that thephase mask be positioned so that the phase of incident light beams canbe shifted a constant amount. For example, the phase mask could bepositioned immediately in front of the data mask so that it is betweenthe data mask and the Fourier transforming lens instead of behind thedata mask as shown in FIG. 1. Other configurations are also possible ifdifferent optical systems are used.

As indicated above, my invention may also be practiced without usingprecisely phase-shifts shifts described above. Speaking generally, myinvention is concerned with the imposition of a substantially constantphase-shift on a significant fraction of a group or array of beams ofradiation that are being Fourier transformed. Only more narrowly is myinvention concerned with the imposition of a phase-shift ofapproximately 180 at random on approximately half of an array of beams.And between theses methods of practicing my invention lie any number ofcompromises. As indicated above, a phase-shift of instead of produceslittle difference in the intensity distribution in the Fourier transformplane. And similarly, little effect is observed if the number of beamsthat are phase-shifted is not exactly 50 percent or if the pattern ofphase-shifts is not purely random. Accordingly, it becomes difficult todefine my invention precisely and one must resort to a definition interms of the result achieved. Specifically, the phase of a large enoughfraction .of beams of radiation is shifted enough and with sufficientrandomness to produce an intensity distribution in the Fourier transformplane that is sufficiently uniform for the intended use.

It is also possible to practice my invention with a phase mask 0 canproduce more than one possible phase-shift. For example, a phase maskcan be used in which approximately one-third of the phase-shiftingsquares shift the phase of an incident beam by approximately 120 andanother third shift the phase by 240 or with a phase mask in whichone-fourth shift the phase 90another fourth 180and another fourth 270.And, in general, if (N-l) is the number of different nonzero phase-shiftto be used, then the phase mask should provide a random array of anapproximately equal number of phaseshifting squares for effectingphaseshifts of each multiple of n 360 /N from n to n (N-l The details offabrication of such a mask will be obvious from the description above ofthe fabrication of the mask of 0 and l 80 phase-shifting squares.

My invention may also be practiced in other contexts than thosedescribed above. While the invention is presently being used for therecording of Fourier transform holograms of ordered arrays of lightbeams such as are formed by the data masks presently used in hologrammemories, use of the invention is by no means so limited. Thus, ifrequired, the invention could readily be used in forming Fouriertransform holograms at frequencies of electromagnetic radiation that aresubstantially different from the frequencies of visible light; andspecifically my invention could be used in acoustic holography which ispracticed at radio frequencies. Of course, the phase mask that is usedmust be able to affect the phase of the electromagnetic radiationpassing through it; and, as is well known, a glass phase mask would notbe suitable at all frequencies of electromagnetic radiation. However,there generally are available equivalent means for producing phaseshifts at different frequencies; and the fabrication and use of anappropriate phase mask for use at a particular frequency will be obviousto one skilled in the art.

It is also apparent that my invention is not limited to Fouriertransform holography but can be practiced wherever a more uniformamplitude or intensity distribution is desired in the Fourier transformof an array of beams of electromagnetic radiation.

What is claimed is:

l. Apparatus for forming a record of the Fourier transform of an arrayof beams of electromagnetic radiation, comprising:

means for forming an array of beams of electromagnetic radiation,

means for shifting the phase of at least one-third of the beams in thearray by an amount that is at least ninety degrees and is substantiallyconstant across each beam, wherein the beams that are phase shifted aredistributed approximately randomly throughout said array and means forrecording the Fourier transform of the beams.

2. The apparatus of claim 1 wherein the probability that a particularbeam is phase shifted ranges from approximately one-third toapproximately two-thirds.

3. The apparatus of claim 1 wherein the amount of phase shift is thesame for all the phase-shifted beams and ranges from approximately toapproximately 220 with respect to the beams that are not phase shifted.

4. The apparatus of claim 3 wherein:

the probability that a particular beam is phase shifted ranges fromapproximately two-fifths to approximately threefifths, the record of theFourier transform is a Fourier transform hologram,

the beams of electromagnetic radiation have a constant phase relationwith a reference beam with which they interfere to form the hologram,and

the beams of electromagnetic radiation are disposed in an ordered array.

5. The apparatus of claim 4 wherein the amount of phase shift isapproximately 180 and the probability that a particular beam is phaseshifted is approximately one-half.

6. The apparatus of claim 4 wherein the means for shifting the phase ofthe beams is a phase mask comprised of an ordered array ofphase-shifting areas that have a one-toone correspondence with theordered array of beams of electromagnetic radiation.

7. The apparatus of claim 6 wherein the phase mask is a glass slide inwhich the phase-shifting areas have been etched.

8. The apparatus of claim 1 wherein: the phase shift in at least some ofthe phase-shifted beams is different from the phase shift in some of theother phaseshifted beams,

the means for shifting the phase of the beams is a phase mask comprisedof an array of phase-shifting areas that have a one-to-onecorrespondence with the array of beams of radiation, there being anapproximately equal number of phase-shifting areas for each of thedifferent phase shifts that are efiected.

9. The apparatus of claim 8 wherein the number of different phase shiftsis (N-l and each of the different phase shifts that can be effected inthe phase-shifted beams of radiation is one of the multiples of n360/Nwhere n ranges from 1 to (N-l

1. Apparatus for forming a record of the Fourier transform of an arrayof beams of electromagnetic radiation, comprising: means for forming anarray of beams of electromagnetic radiation, means for shifting thephase of at least one-third of the beams in the array by an amount thatis at least ninety degrees and is substantially constant across eachbeam, wherein the beams that are phase shifted are distributedapproximately randomly throughout said array and means for recording theFourier transform of the beams.
 2. The apparatus of claim 1 wherein theprobability that a particular beam is phase shifted ranges fromapproximately one-third to approximately two-thirds.
 3. The apparatus ofclaim 1 wherein the amount of phase shift is the same for all thephase-shifted beams and ranges from approximately 140* to approximately220* with respect to the beams that are not phase shifted.
 4. Theapparatus of claim 3 wherein: the probability that a particular beam isphase shifted ranges from approximately two-fifths to approximatelythree-fifths, the record of the Fourier transform is a Fourier transformhologram, the beams of electromagnetic radiation have a constant phaserelation with a reference beam with which they interfere to form thehologram, and the beams of electromagnetic radiation are disposed in anordered array.
 5. The apparatus of claim 4 wherein the amount of phaseshift is approximately 180* and the probability that a particular beamis phase shifted is approximately one-half.
 6. The apparatus of claim 4wherein the means for shifting the phase of the beams is a phase maskcomprised of an ordered array of phase-shifting areas that have aone-to-one correspondence with the ordered array of beams ofelectromagnetic radiation.
 7. The apParatus of claim 6 wherein the phasemask is a glass slide in which the phase-shifting areas have beenetched.
 8. The apparatus of claim 1 wherein: the phase shift in at leastsome of the phase-shifted beams is different from the phase shift insome of the other phase-shifted beams, the means for shifting the phaseof the beams is a phase mask comprised of an array of phase-shiftingareas that have a one-to-one correspondence with the array of beams ofradiation, there being an approximately equal number of phase-shiftingareas for each of the different phase shifts that are effected.
 9. Theapparatus of claim 8 wherein the number of different phase shifts is(N-1) and each of the different phase shifts that can be effected in thephase-shifted beams of radiation is one of the multiples of n360/N wheren ranges from 1 to (N-1).